Question: Solve for $X$. $\left[\begin{array}{rr}4 & -5 \\ 9 & -2 \end{array}\right]+X=\left[\begin{array}{rr}6 & 3 \\ 2 & 11\end{array}\right] $ $X=$
Solution: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}4 & -5 \\ 9 & -2 \end{array}\right]+X=\left[\begin{array}{rr}6 & 3 \\ 2 & 11\end{array}\right]$ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}4 & -5 \\ 9 & -2 \end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}6 & 3 \\ 2 & 11\end{array}\right]$ Then we can rewrite the equation as follows. $A+X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A+X&=B\\\\ X&=B-A\end{aligned}$ Finding $X$ We found that $X=B-A$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=B-A \\\\&=\left[\begin{array}{rr}6 & 3 \\ 2 & 11\end{array}\right]-\left[\begin{array}{rr}4 & -5 \\ 9 & -2 \end{array}\right] \\\\\\&=\left[\begin{array}{rr}(6-4) & (3+5) \\ (2-9) & (11+2) \end{array}\right] \\\\\\&=\left[\begin{array}{rr}2 & 8 \\ -7 & 13\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}2 & 8 \\ -7 & 13\end{array}\right]$